For a universal set U with n(U) = 110, if n(A) = 64, n(B) = 57, and n(A ∩ B) = 29, what is n((A ∪ B)′)?
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Answer and explanation
Correct answer: 18
Use the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 64 + 57 − 29 = 92. The complement of A ∪ B contains all elements of U outside the union. Therefore, n((A ∪ B)′) = n(U) − n(A ∪ B) = 110 − 92 = 18. Thus option A is correct; 92 is the size of the union, not its complement.
Frequently asked questions
What is the correct answer to this question?
18
Why is this the correct answer?
Use the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 64 + 57 − 29 = 92. The complement of A ∪ B contains all elements of U outside the union. Therefore, n((A ∪ B)′) = n(U) − n(A ∪ B) = 110 − 92 = 18. Thus option A is correct; 92 is the size of the union, not its complement.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
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